TY - GEN

T1 - Social welfare and profit maximization from revealed preferences

AU - Ji, Ziwei

AU - Mehta, Ruta

AU - Telgarsky, Matus

N1 - Publisher Copyright:
© Springer Nature Switzerland AG 2018.

PY - 2018

Y1 - 2018

N2 - Consider the seller’s problem of finding optimal prices for her n (divisible) goods when faced with a set of m consumers, given that she can only observe their purchased bundles at posted prices, i.e., revealed preferences. We study both social welfare and profit maximization with revealed preferences. Although social welfare maximization is a seemingly non-convex optimization problem in prices, we show that (i) it can be reduced to a dual convex optimization problem in prices, and (ii) the revealed preferences can be interpreted as supergradients of the concave conjugate of valuation, with which subgradients of the dual function can be computed. We thereby obtain a simple subgradient-based algorithm for strongly concave valuations and convex cost, with query complexity (formula presented), where ϵ is the additive difference between the social welfare induced by our algorithm and the optimum social welfare. We also study social welfare maximization under the online setting, specifically the random permutation model, where consumers arrive one-by-one in a random order. For the case where consumer valuations can be arbitrary continuous functions, we propose a price posting mechanism that achieves an expected social welfare up to an additive factor of (formula Presented) from the maximum social welfare. Finally, for profit maximization (which may be non-convex in simple cases), we give nearly matching upper and lower bounds on the query complexity for separable valuations and cost (i.e., each good can be treated independently).

AB - Consider the seller’s problem of finding optimal prices for her n (divisible) goods when faced with a set of m consumers, given that she can only observe their purchased bundles at posted prices, i.e., revealed preferences. We study both social welfare and profit maximization with revealed preferences. Although social welfare maximization is a seemingly non-convex optimization problem in prices, we show that (i) it can be reduced to a dual convex optimization problem in prices, and (ii) the revealed preferences can be interpreted as supergradients of the concave conjugate of valuation, with which subgradients of the dual function can be computed. We thereby obtain a simple subgradient-based algorithm for strongly concave valuations and convex cost, with query complexity (formula presented), where ϵ is the additive difference between the social welfare induced by our algorithm and the optimum social welfare. We also study social welfare maximization under the online setting, specifically the random permutation model, where consumers arrive one-by-one in a random order. For the case where consumer valuations can be arbitrary continuous functions, we propose a price posting mechanism that achieves an expected social welfare up to an additive factor of (formula Presented) from the maximum social welfare. Finally, for profit maximization (which may be non-convex in simple cases), we give nearly matching upper and lower bounds on the query complexity for separable valuations and cost (i.e., each good can be treated independently).

UR - http://www.scopus.com/inward/record.url?scp=85059002334&partnerID=8YFLogxK

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U2 - 10.1007/978-3-030-04612-5_18

DO - 10.1007/978-3-030-04612-5_18

M3 - Conference contribution

AN - SCOPUS:85059002334

SN - 9783030046118

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 264

EP - 281

BT - Web and Internet Economics - 14th International Conference, WINE 2018, Proceedings

A2 - Harks, Tobias

A2 - Christodoulou, George

PB - Springer Verlag

T2 - 14th International Conference on Web and Internet Economics, WINE 2018

Y2 - 15 December 2018 through 17 December 2018

ER -